As we know, the spectrum
of a time series
has both
a magnitude
and a phase
. The
phase of the spectrum gives information about when the signal
occurred in time. For example, if the phase is predominantly linear
with slope
, then the signal must have a prominent pulse,
onset, or other transient, at time
in the time domain.

For stationary noise signals, the spectral phase is simply
random, and therefore devoid of information. This happens
because stationary noise signals, by definition, cannot have special
``events'' at certain times (other than their usual random
fluctuations). Thus, an important difference between the spectra of
deterministic signals (like sinusoids) and noise signals is that the
concept of phase is meaningless for noise signals. Therefore,
when we Fourier analyze a noise sequence
, we will always
eliminate phase information by working with
in the frequency domain (the squared-magnitude Fourier transform),
where
.